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Title Vectorial wave analysis of dielectric waveguides for optical-integrated circuits using equivalent network approach

Author(s) Koshiba, M.; Suzuki, M.

Citation Journal of Lightwave Technology, 4(6): 656-664

Issue Date 1986-06

Doc URL http://hdl.handle.net/2115/7373

Rights©1986 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material foradvertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists,or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.IEEE, Journal of Lightwave Technology, 4(6), 1986, p656-664

Type article

File Information JLT4_6.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

656 JOURNAL OF ~IGHTWAVE TECHNOLOGY, VOL. LT-4. NO.6, JUNE 1986

Vectorial Wave An'l.lysis of Dielectric Waveguides for Optical-Integrated Circuits Using Equivalent

Network Approach MASANORI KOSHIBA, SENIOR MEMBER, IEEE, AND MICHIO SUZUKI, SENIOR MEMBER, IEEE

~bstract-A vectorial wave analysis of the propagation characteris­tics of open dielectric waveguides for optical-integrated circuits using an equivalent network approach ~s presented. In this approach, all of the contributions from the discrete and continuous parts of the spec­trum and from the TE-TM coupling, which are neglected jn the earlier equ ivalent network approach, are taken into account. To show the va­lidity and usefulness of this formuilition, examples are computed for optical strip waveguides, rib waveguides, rect~nguJar dielectric wave­guides, embossed waveguides, and embedded waveguides.

I. INTRODUCTION

OPEN DIELECTRIC WAVEGUIDES for optical-in­. tegrated circuits as shown in Fig. I, where nj'1 (i =

I, 2 and j = I , 2, 3, 4) is the refractive index and n~) ;;,: n¥) , are very difficult to analyze rigorously. Some analyses of optical strip waveguides, rib waveguides , rectangular dielectric waveguides , embossed waveguides, and embedded waveguides obtained from the somewhat generalized waveguide structure in Fig. I have been at­tempted through the following approaches [I ]: t~e ap­proximate analytical approach [2]-[4]; and the numerical approach [5]-[8] . Although the calculation procedure of the fonner approach is relatively simple, its accuracy is practically insufficient. On the otner hand, the latter ap­proach is more effective for precise investigation. How­ever, the numerical approach is, in general, time consum­ing and expensive . Recently, the equivalent network approach has been developed for open dielectric wave­guides [9]-[16] . The calculation procedure of this ap­proach is relatively simple, and application of the so­called transverse resonance technique yields approximate but fairly accurate analytical expressions for the disper­sion relations for the propagation wavenumber k, in the z­direction. However, since the continuous spectrum con­tributions at the sides of the waveguide are neglected, the equivalent network representation cannot be derived for the waveguides having no discrete modes in region 2 in Fig . I, such as the rectangular dielectric waveguide , the embossed waveguide, and the embedded waveguide [14]. For the analysis of a dielectric waveguide on a ground plane, Koshiba et al. [17], [18] have presented an im-

Manuscript received September 3 , 1985 ; revised January 16 , 1986. The authors are with the Depanment of Electronic Engineering , Hok­

kaido University , Sapporo , 060 Japan. IEEE Log Number 8608007.

Region 2 Region Region 2

W

n(2) , n~') n~2)

/ / / / / /

/ n(2) / y n\1l / n(2) /

, / /

, , / n(2) n(1) / n(2)

/ , / z ,

/ ,

n~2) n~1l nO) ,

Fig I. Dielectric waveguide for optical- integrated circuit s.

proved equivalent network approach in which the contin­uous spectrum contributions are taken into account. How­ever, this approach is based on the scalar wave approximation, in which the coupling produced between TE (E)' == 0) and TM (Hy == 0) modes [8], [12], [13] at the sides of the waveguide is neglected. In [12] and [13], the TE-TM coupling is ta"en into account , but the ap­proximate bounded approach, in which the open dielectric waveguide is enclosed by perfect electric conductors, is used.

This paper presents a more rigorous method of the vec­torial wave analysis of open dielectric waveguides for op­tical-integrated circuits using an equivalent network ap­proach. In this analysis , all of the contributions from the discrete and continuous parts of the spectrum and from the TE-TM coupling are taken into account. To show the validity and usefulness of this formulation, examples are computed for optical strip waveguid!!s, rib waveguides, rectangular dielectric waveguides, embossed waveguides , and embedded waveguides.

II. TRANSMISSION LINE EQUATIONS

For the purpose of the analysis , the waveguide is di­vided into two regions (regions I and 2) as shown in Fig. I , and then, with respect to the x direction, we express the transverse fields [12], [13] in tenns of ali TE (Ey ==

0733-8724/86/0600-0656$01.00 © 1986 IEEE

KOSHIBA AND SUZUKI: ANALYSIS OF DIELECTRIC WAVEGUIDES USING NETWORK APPROACH 657

0) and TM (Hy '" 0) modes [8], [12], [13], as follows:

E?) = [E;oE;iY

. ~ (" (i) fU ) = exp (-jk, z) ,:-, (:: V,q(x) ,q(Y)

+ ~ i,yl V~)(x, p) f~;(Y, p) dP] S ~ O JE;"

(1 )

(2)

where the superscripts i = 1 and 2 denote quantities for regions 1 and 2 i respectively, the subscripts r = 1 and 2 denote quantities for the TE and TM modes, respectively , T denotes a transpose, the summation Eq extends over the discrete modes, p is the wavenumber in the y-directibn of the continuous spectrum inside the medium with the re­fractive index n~o, and ~ ~i) and ~~i ) are given by

s = 0

s = 1,2 (3a)

s = 0

s = 1, 2. (3b)

Here ko is the wavenumber of a vacuum. The radiation mode indicated by s = 0 decays exponentially in the re-

. f f . . d U) ( ). (, ) (, ) [19] glOn 0 re ractIve In ex n4 t2::::; y , Since n3 ~ n4 .

Fat

ko .J(n~)2 - (n~ ) 2 ,; p

there are two independent types of radiation modes [19]. These radiation modes indicated by s = 1 and s = 2 con­sist of standing waves above and below the core region (-I, ,; Y ,; I,) [19]. . '

The mode functions /~(y) and g~;(y) for discrete modes

and the mode functions/~) (y, p) and g~;(y , p) for contin­uous spectrum can be normalized in accordance with the orthonormality statement

Ioo

{I' x f U)(y)}* U) ( ) d x 'q g"q ' Y Y

- 00

(i ) X ix }* . f"q '(y) dy = o".Oqq'

(4a)

Ioo

. x (i ) * - 00 {Ix f,,(Y , p)} . g~.;(y , p') dy

roo (i )

= Loo {g" (y, p) x U* . f~i.;(y , p') dy

= O,,'''(p - p') (4b)

where the asterisk denotes complex conjugate, ix is the unit vector in the x-direction, and Oqq' and ,,(p - p') are the Kronecker" and the Dirac" function" respectively.

The modal voltage V~;(x) and current /~)(x) for discrete modes satisfy the following transmission line equations:

-dVU)(x)/dx = jKU )ZU )/(0(x) rq rq rq rq (Sa)

_ d/(i)( )/dx = . U)yU )VU)( ) rq X JKrq rq rq X (5b)

(i ) .J(kU)' _ k' Krq = rq Z (6a)

r = 1

r = 2 (6b)

(i) (i)

ZU ) = U) _ [K)/WE)q, rq lIYrq - (i)1 (i )

Wl-'2q K2q •

(0 = (kUl/k)2 €lq €O lqO (6c)

U) = (k U) /k)2 I'2q 1'0 'q "U (6d)

where W is the angular frequency, EO and 1'0 are the per­mittivity and the permeability of a vacuum, respectively, and k('Y is the wavenumber of discrete modes for the equiv;{ent layered structure that is uniform along x and z.

On the other hand, the modal voltage V~;(x, p) and cur­rent /~) (x, p) for continuous spectrum satisfy the follow­ing transmission line equations:

-dV~)(x, p)/dx = jK~;(p) Z~;)(p) i~~)(x, p) (7a)

-d/~;(x, p)/dx = jK~;(p) y~)(p) V~;(x, p)

E\i;(p) {kij(n~ ) ' - p'}/w'l'o

I'g;(p) = {kij(n~ ) ' - p'} /W'EO'

(7b)

(8a)

r = 1

r = 2

(8b)

(8c)

(8d)

A summary of the mode functions for discrete modes and continuous spectrum is given in the Appendix.

n( EQUIVALENT NETWORK FOR STEP JUNCTION

The dielectric waveguide as shown in Fig. I will sup­port the propagation of waves having two possible field configurations, classified as the E;q and E~q modes [2] that can be represented by a linear combination of the TE (Ey '" 0) and TM (Hy '" 0) modes [8]. The main field components of the members of the first family are Ex and Hy , while those of the second are Ey and Ex. The sub­scripts p and q indicate the number of extrema of the elec­tric or magnetic field in the x and y directions , respec­tively.

A. E~q Modes

The boundary conditions at each junction plane (x ± W/2) are that both E, and H, are continuous across it.

658 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT·4, NO. 6, JUNE 1986

x= x' ctoss section , we obtain I I I ± (I: V;')(x ' ) r ~ h;' {i, x f ;' )(y)} dy

, = 1 q q J- o:o q

n(1) , n~Zl

y n (I) n(2) jt2 2 2 Z x n(1) n ~Zl t, ,

n~1) n~2)

. {i, x f ;!h p)} dY] dP)

2 .

= I: (I: V;2)(x,) r ~ h; ' {i, x f ;2)(y)} dy r ""' 1 q q J -00 q

Fig . 2 . Step junction hetween two region s.

Let us assume the unkndwn magnetic field in the junction plane x = x ' as shown in Fig . 2 to be h" From (2), the boundary condition for H, gives

(1 2)

The impedances to the -x and + x directions from the junction plane may be expressed as

Applying (4) to (9) , we obtairi

V~(x') = ~ Z,q" /~;(x') (13a)

V(I)(, ) - Z () / (1)( , ) rs X , P - T rs ~ P rs X , P (l3b)

where Z,q" arid Z" ,,(p) are the impedances of discrete modes and continuous spectrum to the 'Fx direction from the junction plane, respectively , and the upper sign (-) refers to i = 1 and the lower sign (+) to i = 2 .

Defining the input impedance Z" .qCx') of the TEq mode (qth TE mode) at x = x' - 0 to the right-hand side in

(9) Fig . 2 as

Z', q(x ' ) = V~~(x ' )/ /~~(x') (14)

and using (lO)! (13) , and (14) , we obtain the following stationary expression:

(lOa) Z", .<x' ) = N;qZ, q+ + Z ( 15)

/~)(x' , p) = r~ {i, x f~h p)}* . h, dy. (lOb) tv'q = I r~ {i, x f f;(y )} * . h,(y) dyj

From (I), the boundary condition E, gives r~ {i, x f~~(Y) }* . h,(y) dyl (l6a)

± ("I: V;'\x') f ;'\y) s m 1 Lq q q

Z = r± ± r r ~ h; (y) L, "lr. 1JJ-oo

2 II! I)

"~ ( I), ( I ) ] + L.. V,, (x , p) f ,,(y , p) dp s=O ! ~ l ) 2 ~ "rl

+ I: (,Z",, (p) {i, x f~) ( Y, p)} s "" 0 ~ ,'

. {i, x f~(y', p)}* dP] . ",(y') dy dyj l

I r~ {i, X f~~(Y)} * . ",(y) d{ (l6b)

if we multiply {i, X (11») by II; and integrate over the In (16b), q' * q for r = I.

KOSHIBA AND SUZUK I: ANALYSIS OF DIELECTRIC WAVEGUIDES US ING NETWORK APPROAC H 659

, ... ,.,::=Jt] (a)

1 : N2q

YI • . , (x,,::JID OJ (b)

Fig. 3. Equivalent network for the step junction in Fig. 2. (a) E~ modes. (b) E;;" modes.

Relation (15) can be interpreted in tenns of a simple equivalent network as shown in Fig. 3(a). In the scalar wave approximation , the summation 1:;:1 in (l6b) is re­moved and r is replaced by I .

B. E~q M odes

Referring to (9)-(16) , we obtain

Yi, .q(x') = N~q Y2q+ + Y (17)

N2q = I r~ {g;~(y) X i, }* . e,(y) dY/

r~ {g;~(y) x ix }* . e,(y) dyl (18a)

Y = L~I '~I ) r~ e;(y)

. [~ Y,q,~ { g~J.(y) X ix} { g~~,(y') X i, }*

. { g~)(y', p) X ix }* dPJ . e,(y') dy dY'3/ I r~ { g;~(y) X ix}* . e,(y) dYI' (l8b)

where e, is the unknown electric field in the junction plane x = x' in Fig . 2, Yi, .ix') is the input admittance of the TMq mode (qth TM mode) at x = x' - 0 to the right-hand side in Fig . 2, and Y,q~ and Y,,~( p) are the admittances of discrete modes and continuous spectrum to the 'Fx di­rection from the junction plane , respectively. In (18b), q' * q for r = 2.

Relation (17) can be interpreted in tenns of a simple equivalent network , as shown in Fig . 3(b). In the scalar wave approximation , the summation 1:;: 1 in (18b) is re­moved and r is replaced by 2.

i--w--J (a)

i--w--J (b)

Fig. 4 . Equivalent network for the dielec tric waveguide in Fig. I . (a) E;'" modes. (b) Ef,., modes.

IV . DISPERSION RELA nONS

Taking all of the contributions from the di screte and continuous parts of the spectrum and from the TE-TM coupling into account , we obtain the final equivalent net­works as shown in Figs. 4(a) and (b) for the E;q and p"q modes in the original structure (Fig. I) , respectively. To evaluate the turns ratio N,q (r = I , 2) of the transfonner, the series impedance Z, and the shunt admittance Y ex­plicitly , we assume the fonn for the field h, or e, in the junction plane as

h = a( l)g( I)(y) t Iq Iq

+ a (2)g(2)(y) I q Iq , E~q modes (19a)

e = a (l)j (I)(y) / 2q 2q

+ a (2)j (2)( y) 2q 2q • E~q modes. (19b)

Using the Ray leigh- Ritz technique and ~4), the ratio of variational parameters in (19), namely a~;/a~~) == a~~, is

calculated by the fallowing condition:

azl" . .<x' ) /aa ~J = 0,

aYi" . .<x' ) /aa ;~ = 0,

E;q modes

E~q modes.

(20a)

(20b)

The value of a~~ allows the detennination of the turns ratio, the series impedance, and the shunt admittance in Fig. 4.

For the waveguide having no discrete modes in region 2{ we assume the fonn for the field h, or e, to be h, = gl~(Y) or e, = j;~(y) , respectively. In this case, N 1q =

N2q = 0, and Z and Y can be calculated directly from (16b) and (l8b) without using (20), respectively.

A. E;q M odes

From the symmetry of the structure (Fig. I ), we rec­ognize that two classes of solutions are possible: those (p = I , 3, 5, ... ) for which the main field components of the E;q modes, namely Ex and Hy, are symmetric about the midplane (x = 0) and those (p = 2, 4 , 6, ... ) for which they are anti symmetric. The symmetric and anti ­symmetric modes then correspond to the free resonances of the short-circuit and open-circuit bisections of the equivalent network , respectively [14) .

From the transverse resonance condition [9)- [ 18], we

660

find the dispersion relation for the wavenumber k, to be

(2 1)

where Z and Z are the input impedances seen looking in opposite directions at x = WI2 - 0 in the equivalent net­work.

B. E~q Modes

For the E~ modes , the symmetric and antisymmetric modes correspond to the free resonances of the open-cir­cuit and short-circuit bisections of the equivalent net­work, respectively [14].

The dispersion relation is given by

(22)

where Yand Yare the input admittances seen looking in opposite directions at x = WI2 - 0 in the equivalent net­work.

v. COMPUTED R ESULTS

Fig. 5 shows dispersion characteristics for the E~q modes of the rib waveguide . Solid lines are the results of the vectorial wave ana lysis . Dashed lines are the results of the scalar wave analysis in [14]. Our results indicated by so lid lines agree well with the results of the vectorial wave analysis using the mode-matching method [8]. The accuracy of the scalar wave analysis [14] becomes poorer for the higher-order modes.

Figs. 6(a) and (b) show dispersion characteristics for the Ei~1 and E~q modes of the rectangular dielectric wave­guide, respectively , where the normalized frequency vand the normalized guide index b are given by

v = kol J"l - "lhr b = {(k/ ko)' - "ll/(nl - "i).

(23)

(24)

Solid lines are the results of the vectorial wave analysis. Dashed lines are the results of the scalar wave analysis in [1 7] and [18]. Effects of the TE-TM coupling are larger for the higher-order modes (E~ , and E~ , modes) than for the fundamenta l modes (£;, and E{, modes). Our results indicated by sol id lines agree well with the results of the vectorial wave analysis using the collocation method [5].

Figs. 7 and 8 show dispersion characterist ics of the em­bossed waveguide and of the embedded waveguide, re­spectively (vectorial wave analysis). Solid and dashed lines are for the E~q and E~ modes , respectively . Com­parison of our results with the results of the vectorial wave analysis using the finite-element method [7] shows good agreement.

Table I shows the numerical results for the E;q modes of the optical strip waveguide, where ,,~' ) = n ~') == '" = Fs ,,(') = 1,(2) = ,,(') == " = J2 375 n(l) = ,,(2) =

"3 3 2 2 "4 4 II;') = 1.0, W = 81" and I, = I, == 21. In Table I, the results of Marcatili ' s method [4], the effective index method [3], the variational method [6], and the finite-e1e-

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT·4, NO.6, JUNE 1986

' .74

'.73

' .72 . ~

~

1-71

' .10

E:, modt>

E i, modI!'

ord~, TE mod~ 01 ,h" uniform sl,.b W&vlf9Uide

n,

",,,1 .742, n,=1.69, ", =1·0

12.1,. W",6t,

Fig . 5. Di spersion characteristics of Ihe rib waveguide (veclOrial wave analysis, solid lines; scalar wave analysis 1141. dashed lines).

,.Or---- ------ ---- -----, • Goell

0·5

r-w, ~

[JJ 7 ~ n, 7 7 7 7

7 ~ 7 ,

n, ::1 · 5 E ~, "' .... E;, n2:1.O "/

/

W=t / /

/ /

/

' ·0 ' ·0 3·0 , (a)

'.0,------------------,

0·5

" , =1 ·5

"2=1 .0

W=I

, (b)

Fig. 6. Dispersion characteristics of the rectangu lar die lectric waveguide (vec torial wave analysis , solid lines; scalar wave analysis 11 71. 11 81: dashed lines). (a) E~ modes. (b) E~ modes.

ment method [7] are also presented . Our results agree well with the results of the vectorial wave analyses using the variational method and the finite-element method. Our re­sults obtained by neglecting the TE-TM coupling (scalar wave approximation) are given in the parentheses . Effects of the TE- TM coupling on the dispersion characteristiCS of this waveguide are very small , because the optical strip waveguide is formed by the dielectric strip that perturbs

KOS HIB A AND SUZUKI: ANALYSIS OF DIELECTRIC WAVEGU IDES USING NETWORK APPROACH 661

D

\.0,----------------------;

0 ·5

Veh, Ha, Dong. Brown

E:, ---/'

Erl E i 1 ---11---1, El,

(a)

;---w ------,

"3 1 ", I: n,

" , ='·50 " 2 ='·45

I'll =1 ·0

W=41

\.0,------------------ ---,

• YE'h, Ha, Dong. Brown

05

, (b)

n , d ·50

" 2 ='·45

",='.0 W=2t

\·0,---------------------,

05

• YE"h, Ha, Dong. Brown

n,

1'1 , =' ·50

"2 =' ·45 "]:1 .0

W=I

, • /. ,

Er, / E:, , , 0, 0' I

, (c)

/. /.

/. /.

/.

, , ,

/. /.

/

El, / E:, /

/ , ,

/. /.

Fig. 7. Dispe rsion characteristics of the embossed waveguide (£~ modes, solid li nes; E~ modes, dashed lines). (a) W = 41 . (b) W = 2r. (c) W = I.

the planar dielectric waveguide in which TE and TM modes are uncoupled.

VI. C ONCLUSIONS

Vectorial wave analysis of propagation characteristics of open dielectric waveguides for optical-integrated cir­cuits has been perfonned by using an equivalent network approach. In this approach, all of the contributions from the discrete and continuous spectrum and from the TE­TM coupling, which are neglected in the earlier equiva­lent network approach , are taken into account, and there-

\.0.-------------- - -------,

0· 5

i -----;-,....

, - ............. / / , /

/. /

----

/ / n l r--- w~ ,,' --'-ir------+-. // // -n2 I ", I~

-----,(" , , , ---H , , , --{H' , ,

\ ·0 , (a)

Yf'h, Ha , Dong. Br own

, (b)

,·0

n , =' ·50

"2 =\·45

"]='.0 W=2t

' ·0

\.0,----------------------,

n,

(>5

V /

E:, / q , 1'1, ='·50 '/ n l =1 ·45 I

, "J:1·O /

I /

V

I " I

W:d I El, OL-~~-~~/L-L-~~~~~~L_~-~~~~ o ,·0 2·0 ) ·0 ,

(c)

Fig. 8. Dispersion characteri stics of the embedded waveguide (£~ modes, solid lines; £~ modes, dashed lines). (a) W = 4/. (b) W = 21. (c) W = I.

fore the present equivalent network approach can give more accurate results for the various dielectric wave­guides over a wide range of frequencies.

ApPENDIX

The mode functions for discrete modes are

r = r = 2

(AI)

662

Mode

E~ 1

E~l

JOURNA L OF LIGHTWAV E TECHNOLOGY, VOL. LT-4, NO.6, JUNE 1986

TABLE! D ISPERSION C HARACTER ISTICS FOR THE E~ M ODES OF THE O PTICAL S TRIP

W AVEGUIDE

Norma l ized frequency

•

0.25

0 .63

0 . 25

0.63

Marcatili's method

0.167

0.716

0.042

0.695

r = I

r = 2

Normalized guide index b

Effecti ve Variational Finite index

method el eme nt Present analysis method method

0.278 0.2590 0.270 0 .2675 (0 . 2674)

0 .725 0 .7226 0.724 0 . 7242 (0 . 7242)

0 .1 78 0.1313 0. 1476 (0 . 1470)

0,706 0.6995 0. 7036 (0 . 7035)

(A2) B,q. , Cfq , 1 - (m,krq , I / m 3(X. rq ,3)Srq, 1

(mlkrq. l/m3arq,3 ) Crq, ] + Srq, I (A7)

where for simplicity the superscript i is abbreviated , r/>;q(Y) = dr/>,q(y )/dy , and m( y) and r/> ,q(Y) are given by B,q,'

Crq ,2 - (m2krq,2/m4C(rq ,4)Srq,2

(m, k,q,,/m4a ,q.4) C,q,2 + S,q,' (A8)

Brq ,3 = C rq. 1 - B rq. 1Srq, 1 (A9)

m( y) - t , :5 Y :5 0

o :5 Y :5 t, (A3)

A rq,4 = C rq•2 + B rq ,2 S rq,2 (AlO)

Here

m"

I

B,q,3 exp {a,q.3(y + t, )},

Y :5 -t,

cos krq ,lY + Brq , I sin krq,1Y,

-t, :5 y:5 0 (A4)

Cirq,j .. Jk;q - k6nJ, k rq,j ~k ~ n} - k ;q,

Srq,j = sin k rq./ j ,

C rq,j = cos krq.j tj ,

j 3, 4 (All )

j I , 2 (AI 2)

j I , 2 (AI 3)

j = I , 2 . (AI4)

~ cos krq ,2Y + Brq ,2 sin krq . 2y ,

o :5 y :5 t, The dispersion relation for the wavenumber k,q is given by

A,q.4 exp { - a ,q.4(Y - t, )},

t, :5 y, The mode functions for continuous spectrum are

j ,,(y, p)

g,,( y, p)

[[0 r/>, ,(y, p)]T,

[jm(y) r/> ,,(y , p) m( y) r/>,,(y, p) k,l (k ~ nl - P' )f,

[[ -r/>,,(y, p) j r/>:,(y, p) k,l(k6nl - p2)1', r = I

[0 j r/>,,(y, p)l', r = 2

where

r = I

r = 2

r/>;,(y, p) = dr/> ,,(y, p)/dy .

(AIS)

(AI6)

(A17)

[I , r=

j = I , 2, 3, 4 , When s

(AS ) by 0(0 :5 P :5 ko ../nl - nil , r/> ,,( y, p) is given

lin},

, + .2.; mj {tj + S,q.j C,q.j/k,q.j

J=l

r = 2

+ B;q./tj - S,q,j C,q./k,q,j ) + 2B,q.j S;q,/k,q.j }]

(A6)

I r/>,,(y, p) = r;:;:­

v D rs

A".J cos pry + t,) + B".J sin pr y + til ,

cos O"l Y + Brs , 1 sin O" lY,

cos 0"2Y + B rs ,2 sin 0"2Y,

A".4 exp { - ~(y - t, )},

Y :5 - t,

- t, :5 Y :5 0

0 :5 Y :5 t,

t, :5 Y

(AI8)

KOSHIBA AND SUZUKI: ANALYSIS OF DIELECTRIC WAVEGUIDES USING NETWORK APPROACH 663

Here

aj

Sj

Cj

D"

(m,u,lmlal)B""

-{C, - (m,a,lm471)S, }1

{(m,a,lm471)C, + S,)

(mla l/m3P) (SI + B",ICI)

C, + B"" S,

..Jk6(ni - n~) _ P'

..Jk6(nJ - ni) + p2, j

sin a/j, j

cos aj l} , j

I , 2

I , 2

I , 2,

(AI9)

(A20)

[X + ..JX' + I,

B -",' - X - -.ix' + I,

X = XNIXo

s

s = 2 (A39)

(A40)

XN = m3[(mlal/m3P)' {(m, a,lm,a,)'Ci - sil (A21)

(A22)

(A23)

(A24)

ci + (m, a,lmlal)' Si] + m4(t1lp) [{ (m,a,lm4t1)' I} (Cl - Sl)]

(A41)

(A25) + m4(t1Ip) {I - (m, a, iln4t1)' } S, C, ] (A42)

(A26) for n I :S n"

(A27) ACKNOWLEDGMENT

(A28) The authors wish to thank H, Ishii and T, Hayashi for

When s = I and 2 (ko "n""'3-----,' ) '" ( )" their assistance in numerical computations, y n4 :S P , '1'" , y, P IS given

by

A",3 cos pry + t l) + B",3 sin pry + t l)' Y :5 - II

cos alY + BrL 1 sin a lY, -t l :S Y :S 0 </>,,(y , p)

.fD" cos a2Y + Brs .2 sin 0"2Y, 0 (A29)

:S Y :S t,

A",4 cos t1(y - I,) + B",4 sin t1(y - I,), t 2 $. y,

Here

D" = (7f/2) {m3(A ;", + B ;",l

+ m4(t1lp) (A;,,4 + B ;,,4)} (A30)

BrsA (m,a,lm4t1) (- S, + B"" C, ) (A3 i)

t1 = ..Jk6(n~ - ni) + p' (A32)

and B", 3, A n ,3, and A n ,4 are the same as (A22), (A23) , and (A24), respectively , B",I and Bn " are given by

= [X + -.ix' + I, X - -.ix' + I ,

X = XNIXo

s

s 2

XN = m3[{(rn lallm3P)' - I} (Ci - sill

(A33)

(A34)

(A35)

+ m4(t1lp) [(m,a,lm4t1)' {(mlal /m, a,)' C l - sll - cl + (mlal/m,a,)' sl] (A36)

Xo 2[rn3{(mla l/m3 P)' - I}SICI

+ m4(t1lp) (mlal/m,a,) {l - (m, a, lm4t1 ),}S, C, ]

(A37)

for n[ ;::: n2, and

Iirs, I = (m20"2Imlal)Brs.2 (A38)

REFERENCES

[I] M. J. Adams. An Introduction to Oprical Wa veguides. New York: Wiley - Interscience, 198 1.

12] E. A. 1. Marcatili , "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell. Syst. Tech. 1., vol. 48, pp. 207 1-2102, Sept. 1969,

(3] H. Furuta , H. Noda , and A. Ihaya , "Novel optical waveguide for integrated optics, " Appl. Opt., vol. 13, pp . 322-326, Feb. 1974.

14) E. A. 1. Marcatili , "Slab-coupled waveguides," Bell Syst. Tech. J. , vol. 53, pp . 645-674 , Apr. 1974 .

15] 1. E. Ooell , "A circular-harmonic computer analysis of rectangular dielectric waveguides, " Bell Syst. Tech. J . , vol. 48 , pp. 2133-2160, Sept . 1969.

[6] M. Ohlaka , M. Matsuhara, and N. Kumagai , "Analysis of the guided modes in slab-coupled waveguides using a variational method, " IEEE J. Quantum Electrqn. , vol. QE-12, pp. 378-382, Jul y 1976.

17] C. Yeh, K. Ha , S. B. Dong, and W. P. Brown , " Single-mode optical waveguides," Appl. Opt., vol. 18 , pp. 1490- 1504, May 1979.

[8] K. Yasuura, K. Shimohara, and T. Miyamoto , "Numerical anal ysis of a thin-fi lm wavegu ide by mode-matching method ," J. Opr. Soc. Amer., vo l. 70, pp. 183-191, Feb. 1980.

19) M. Koshiba and M. Suzuki, "Microwave network analyses of dielec­tric waveguides for mill imeter waves made of die lectric strip and planar dielectric layer," Trans. In st. Electron. Commun. Eng. Japan, vol. E63 , pp. 344-350, May 1980.

[10] M. Koshiba and M. Suzuki, "Equivalent network anal ys is of dielec­tric thin-film waveguides for optical-integrated circuits," TrailS. Illst. Electron. Commun, Elig. Japan , vol. E64 , pp . 266- 267 , Apr. 1981.

I ll] M. Koshi ba and M. Suzuki , " Equivalent network analysis of dielec­tric thin -film waveguide with trapezoidal cross seclion," Electron. Lett. , vol. 17 , pp. 283-285, Apr. 1981.

[121 S. T. Peng and A. A. Oliner, " Guidance and leakage propenies of a class of open dielectric waveguides: Pan I- Mathematical fonnula­lion ," IEEE Trans. Microwave Theory Tech., vol. MTT-29, pp. 843-855, Sept. 1981.

[\3] A. A. OHner, S. T . Peng, T . i. Hsu , and A. Sauchez, " Guidance and leakage propenies of a class of open dielectric waveguides: Pan U-

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New physical effects ." IEEE Trails. Microwa ve Th eory Tech . , vol. MTT-29 , pp. 855-869, Sept. 1981.

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[IS] N. DagJi and C . G. Fonstad, "Analysis of rib waveguides with sloped rib sides," Appl. Phys. Letl .. vol. 46, pp. 529-53 \ , Mar. 1985.

[\6] N. Dagli and C . G. Fonstad, "Analysis of rib dielectric wave­guides," IEEE J. Quanrul11 Electrol/ . , vol. QE-21 , pp. 315-321, Apr. 1985.

(1 7] M. Koshiba, H. Ishii, and M. Suzuki, " Improved equivalent network analysis of a dielectric waveguide placed on a ground plane ," TrailS . inst. Electron. Commull. Eng. Japan , vol. E65, pp. 572-578, Oct. 1982 ,

[18) M. Koshiba , H . Ishii, and M. Suzuki , " Simple equivalent network for a rectangular dielectric image guide," Electron. Lett., vol. 18, pp. 473-474 , May 1982.

[19] D. Marcuse , Theory of Dielectric Optical Waveguides. New York: Academic , 1974 .

* Masanori Koshiba (SM '84) was born in Sap­poro , Japan, on November 23, 1948. He received the B.S. , M.S ., and Ph.D. degrees in electronic engineering from Hokkaido University , Sappbro, Japan, in 1971, 1973. and 1976, respectively.

In 1976, he joined the Depanment of Elec­tronic Ertgineering , Kitami In stitute of Technol­ogy , Kitami, Japan. Since 1979, he has been an Assistant Professor of Electronic Engineering at Hokkaido University . He has bee n engaged in re­search on surface acoustic waves , dielectric opti-

JOURNAL OF LIGHTWAVE TECHNOLdOY, VOL. LT-4, NO.6, JUNE 1986

cal waveguides, and applications of finite -e lement and boundary-element methods to field problems .

Dr. Koshiba is a member of the Institute of Electronics and Communi_ cation Engineers of Jap:'! n . the Institute of Te levision Engineers of Japan, the In stitute of Electrical Engineers of Japan, the Japa n Society fo r Simu_ lation Technology , arid the Japan Society for Computational Methods in Engineering.

*

Michio Suzuki (SM'57) was bo rn in Sapporo, Ja-

0, pan, on November 14,1923 . He received the B. S. and Ph.D. degrees in electrical engineering from Hokkaido University, Sapporo , Japan , in 1946 and 1960, respectively.

From 1948 to 1962, he was an Assistant Pro­fessor of Electrical Engineering at Hokkaido Uni­versity. Since 1962 , he has been a Professor of Electronic Engineering at Hokkaido University. From 1956 to 1957, he was a Research Associate at the Microwave Research Institute of Po lytech­

nic Institute of Brooklyn, Brookly n , NY. Dr. Suzuki is a member of the Institute of Electronics and Communi­

cation Engineers of Japan , the Institute of Electrical Engineers of Japan , the Institute of Television Engineers of Japan , the Japan Society of In for­matio n and Communication Research, and the Japan Society for Simulation Technology.